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Colloquium: Simulating non-Markovian dynamics in open quantum systems

Meng Xu*, Vasilii Vadimov†, J. T. Stockburger‡, and J. Ankerhold§

Meng Xu*

  • Institute for Complex Quantum Systems and IQST, Ulm University, Albert-Einstein-Allee 11, 89069 Ulm, Germany

Vasilii Vadimov†

  • QCD Labs, QTF Centre of Excellence, Department of Applied Physics, Aalto University, P.O. Box 15100, 00076 Aalto, Finland

J. T. Stockburger‡ and J. Ankerhold§

  • Institute for Complex Quantum Systems and IQST, Ulm University, Albert-Einstein-Allee 11, 89069 Ulm, Germany

  • *Contact author: meng.xu@uni-ulm.de
  • †Contact author: vasilii.1.vadimov@aalto.fi
  • ‡Contact author: juergen.stockburger@uni-ulm.de
  • §Contact author: joachim.ankerhold@uni-ulm.de
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Rev. Mod. Phys. 98, 021002 – Published 11 May, 2026

DOI: https://doi.org/10.1103/w3nw-hbjc

Abstract

Recent advances in quantum technologies and related experiments have created a need for highly accurate, versatile, and computationally efficient simulation techniques for the dynamics of open quantum systems. Long-lived correlation effects (non-Markovianity), system-environment hybridization, and the necessity for accuracy beyond the Born-Markov approximation form particular challenges. Approaches to meet these challenges, originating from different fields such as hierarchical equations of motion, Lindblad-pseudomode formulas, chain-mapping approaches, quantum Brownian motion master equations, stochastic unravelings, and refined quantum master equations, have been introduced. This diversity, while indicative of the field’s relevance, has inadvertently led to a fragmentation that hinders cohesive advances and their effective cross-community application to current problems for complex systems. How are different approaches related to each other? What are their strengths and limitations? A systematic overview and a concise discussion are presented addressing these questions. This Colloquium makes use of a unified framework that conveniently allows different schemes to be linked and in turn may also catalyze further progress. In line with the state of the art, this framework is formulated not in a fully reduced space of the system but in an extended state space that, in a minimal fashion, includes effective reservoir modes. It offers a comprehensive understanding of existing methods, elucidating their physical interpretations, interconnections, and applicability.

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Rev. Mod. Phys. 98, 021002– Published 11 May, 2026
  • Received 24 March 2024
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